Variety of Idempotents in Nonassociative Algebras

  • Yakov KrasnovEmail author
  • Vladimir G. Tkachev
Part of the Trends in Mathematics book series (TM)


In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least n2 − 1 nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension n. We also discuss the exceptionality of the eigenvalue \(\lambda =\frac 12\) which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrized algebras.


Idempotents Nonassociative algebras Metrized algebras Peirce spectrum Axial algebras 



V. Tkachev has been partially supported by Stiftelsen GS Magnusons fond, grant MG2018-0042. The authors are very grateful to the reviewer for his/her careful and reading of the paper.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of MathematicsLinköping UniversityLinköpingSweden

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